As per this model, there will be two simultaneous operating investment schemes namely long term (fixed) investment and short term (operational) investment. There are two types of complementary growth portfolios namely risky assets and safety assets, operates within each of the fixed and operational investments. Hence this study describes and developed a stochastic model with Bivariate Poisson linear growth and loss processes with two-dimensional investment schemes. The proposed process has the combination of four possible investments such as (i) Fixed Investments with Risky Assets, (ii) Fixed Investments with Safety Assets, (iii) Operational Investments with Risky Assets and (iv) Operational Investments with Safety Assets. Two schemes of investments are connected by a linear combination for getting the joint effect of the total investment policy.
This study has addressed a two-dimensional investment stochastic model with bivariate risk management and portfolio diversification within each dimension. The activities like bonus shares, liquidation, transfer of investments from risky assets to safety assets and vice-versa are expressed with the combination of fixed and operational investments. Each bivariate process is influenced with initial fixed investments in risky and safety assets; as well as initial operational investments in risky and safety assets.
Usually, the objective of Portfolio diversification is to minimize the risk in the investments. However, continuous diversification without any absence makes some times the Portfolio is towards for loss in the long run. The logic of jointing the total effect of portfolio diversification with fixed investment and operational investment through a linear combination of binary variables is considered in this study. Further, it is assumed that the Poisson processes of diversification during the fixed investment and operational investment are complementary. However, the model development is designed to study the factors on growth, evolutions and stipulations and loss processes as an overall phenomenon by combining the conditions of risky and safety assets.
Events will be occurred in non-overlapping intervals of time and are statistically independent. a, b, c, d, e, f, g and h be binary constants such that a, b, c, d, e, f, g, h = (0, 1). The notations to use in the formulation are as follows.
i. mfr: fixed investments in risky assets
ii. mor: operational investments in risky assets
iii. nfs: fixed investments in safety assets
iv. nos: operational investments in safety assets
v. αifr: Growth Rate of initial fixed investments in risky assets per unit time
vi. αifs: Growth Rate of initial fixed investments in safety assets per unit time
vii. αior: Growth Rate of initial operational investments in risky assets per unit time
viii. αios: Growth Rate of initial operational investments in safety assets per unit time
ix. αfr: Growth Rate of fixed investments in risky assets per unit time from the existing mfr
x. αfs: Growth Rate of fixed investments in safety assets per unit time from the existing nfs
xi. αor: Growth Rate of operational investments in risky assets per unit time from the existing operational investments in risky assets
xii. αos: Growth Rate of operational investments in safety assets per unit time from the existing operational investments in safety assets
xiii. δor: Loss Rate of operational investments in risky assets per unit time from the existing operational investments in risky assets
xiv. δos: Loss Rate of operational investments in safety assets per unit time from the existing operational investments in safety assets
xv. δfr: Loss Rate of fixed investments in risky assets per unit time from the existing fixed investments in risky assets
xvi. δfs: Loss Rate of fixed investments in safety assets per unit time from the existing fixed investments in safety assets
xvii. τor: Transition Rate of operational investments from risky assets to safety assets per unit time from the existing operational investments in risky assets
xviii. τos: Transition Rate of operational investments from safety assets to risky assets per unit time from the existing operational investments in safety assets
xix. τfr: Transition Rate of fixed investments from risky assets to safety assets per unit time from the existing fixed investments in risky assets
xx. τfs: Transition Rate of fixed investments from safety assets to risky assets per unit time from the existing fixed investments in safety assets
i) arrival of a share to the group of Risky Assets during Δ t through immigration from external sources provided there exits “ l ” number of shares at time “t” is l { a α i f r + ( 1 − a ) α i o r } Δ t + o ( Δ t ) ;
ii) arrival of share to the group of Safety Assets during Δ t through immigration from external sources provided there exits “ l ” number of shares at time “t” is l { b α i f s + ( 1 − b ) α i o s } Δ t + o ( Δ t ) ;
iii) growth in investment within the group of Risky Assets, during Δ t provided there exits “mfr” number of shares in fixed investments and “mor” number of shares in operational investments already in the said group at time “t” is { c m f r α f r + ( 1 − c ) m o r α o r } Δ t + o ( Δ t ) ;
iv) growth in investment within the group of Safety Assets, during Δ t provided there exits “nfs” number of shares in fixed investments and “nos” number of shares in operational investments already in the said group at time “t” is { d n f s α f s + ( 1 − d ) n o s α o s } Δ t + o ( Δ t ) ;
v) Transition of share from the group of Risky Assets to the group of Safety Assets during Δ t provided there exits “mfr” number of shares in fixed investments and “mor” number of shares in operational investments are in the Risky Assets group at time “t” is { e m f r τ f r + ( 1 − e ) m o r τ o r } Δ t + o ( Δ t ) ;
vi) Transition of share from the group of Safety Assets to the group Risky Assets of during Δ t provided there exits “nfs” number of shares in fixed investments and “nos” number of shares in operational investments in Safety Assets group at time “t” is { f n f s τ f s + ( 1 − f ) n o s τ o s } Δ t + o ( Δ t ) ;
vii) Withdrawal of share (Liquidation) from the group of Risky Assets during Δ t provided there exits “mfr” number of shares in fixed investments and “mor” number of shares in operational investments in the said group at time “t” is { g m f r δ f r + ( 1 − g ) m o r δ o r } Δ t + o ( Δ t ) ;
viii) Withdrawal of share (Liquidation) from the group of Safety Assets during Δ t provided there exits “nfs” number of shares in fixed investments and “nos” number of shares in operational investments in the said group at time “t” is { h n f s δ f s + ( 1 − h ) n o s δ o s } Δ t + o ( Δ t ) ;
ix) No growth in number of shares to the groups of Risky Assets and Safety Assets from external sources, no growth in investment within the group of Risky Assets and Safety Assets, no transformation of shares one group to the other group, no withdrawal of shares from the groups of Risky Assets and Safety Assets during an infinitesimal interval of time Δ t is
1 − { l [ ( a α i f r + ( 1 − a ) α i o r ) + ( b α i f s + ( 1 − b ) α i o s ) ] + m f r ( c α f r + e τ f r + g δ f r ) + m o r [ ( 1 − c ) α o r + ( 1 − e ) τ o r + ( 1 − g ) δ o r ] + n f s ( d α f s + f τ f s + h δ f s ) + n o s [ ( 1 − d ) α o s + ( 1 − f ) τ o s + ( 1 − h ) δ o s ] } ⋅ Δ t + O ( Δ t )
x) Occurrence of other than the above events during an infinitesimal interval of time Δ t is O ( Δ t ) 2 .
Let Pm,n(t) be the probability that there exists “m” shares in Risky Assets group and “n” shares in Safety Assets at time “t”. Further let P m , n ( t + Δ t ) be the probability that there exists “m” shares in Risky Assets group and “n” shares in Safety Assets at time “t + Δt”, it may happened as the processes of
p m , n ( t + Δ t ) = p m , n ( t ) p 0 , 0 ( Δ t ) + p m , n − 1 ( t ) p 0 , 1 ( Δ t ) + p m − 1 , n ( t ) p 1 , 0 ( Δ t ) + p m , n + 1 ( t ) p 0 , − 1 ( Δ t ) + p m + 1 , n ( t ) p − 1 , 0 ( Δ t ) + p m + 1 , n − 1 ( t ) p − 1 , + 1 ( Δ t ) + p m − 1 , n + 1 ( t ) p + 1 , − 1 ( Δ t ) + o ( Δ t ) 2
P ′ m , n = − { l [ ( a α i f r + ( 1 − a ) α i o r ) + ( b α i f s + ( 1 − b ) α i o s ) ] + m [ ( c α f r + ( 1 − c ) α o r ) + ( e τ f r + ( 1 − e ) τ o r ) + ( g δ f r + ( 1 − g ) δ o r ) ] + n [ ( d α f s + ( 1 − d ) α o s ) + ( f τ f s + ( 1 − f ) τ o s ) + ( h δ f s + ( 1 − h ) δ o s ) ] } P m , n ( t ) + [ l ( b α i f s + ( 1 − b ) α i o s ) + ( n − 1 ) ( d α f s + ( 1 − d ) α o s ) ] P m , n − 1 ( t ) + [ l ( a α i f r + ( 1 − a ) α i o r ) + ( m − 1 ) ( c α f r + ( 1 − c ) α o r ) ] P m − 1 , n ( t ) + [ ( n + 1 ) ( h δ f s + ( 1 − h ) δ o s ) ] P m , n + 1 ( t ) + [ ( m + 1 ) ( g δ f r + ( 1 − g ) δ o r ) ] P m + 1 , n ( t ) + [ ( n + 1 ) ( f τ f s + ( 1 − f ) τ o s ) ] P m − 1 , n + 1 ( t ) + [ ( m + 1 ) ( e τ f r + ( 1 − e ) τ o r ) ] P m + 1 , n − 1 ( t ) ; for m , n ≥ 1 (2.4.1)
P ′ 0 , 0 ( t ) = − { l [ ( a α i f r + ( 1 − a ) α i o r ) + ( b α i f s + ( 1 − b ) α i o s ) ] } P 0 , 0 ( t ) + ( h δ f s + ( 1 − h ) δ o s ) ⋅ P 0 , 1 ( t ) + ( g δ f r + ( 1 − g ) δ o r ) ⋅ P 1 , 0 ( t ) (2.4.2)
P ′ 1 , 0 ( t ) = − { l [ ( a α i f r + ( 1 − a ) α i o r ) + ( b α i f s + ( 1 − b ) α i o s ) ] + [ ( c α f r + ( 1 − c ) α o r ) + ( e τ f r + ( 1 − e ) τ o r ) + ( g δ f r + ( 1 − g ) δ o r ) ] } P 1 , 0 ( t ) + ( h δ f s + ( 1 − h ) δ o s ) P 1 , 1 ( t ) + ( g δ f r + ( 1 − g ) δ o r ) P 2 , 0 ( t ) + ( f τ f s + ( 1 − f ) τ o s ) P 0 , 1 ( t ) (2.4.3)
P ′ 0 , 1 ( t ) = − { l [ ( a α i f r + ( 1 − a ) α i o r ) + ( b α i f s + ( 1 − b ) α i o s ) ] + [ ( d α f s + ( 1 − d ) α o s ) + ( f τ f s + ( 1 − f ) τ o s ) + ( h δ f s + ( 1 − h ) δ o s ) ] } P 0 , 1 ( t ) + ( g δ f r + ( 1 − g ) δ o r ) P 1 , 1 ( t ) + ( e τ f r + ( 1 − e ) τ o r ) P 1 , 0 ( t ) + 2 ( h δ f s + ( 1 − h ) δ o s ) P 0 , 2 ( t ) (2.4.4)
With the initial condition
P M o N o ( t ) = 1 , P i , j ( 0 ) = 0 , ∀ i ≠ M o , j ≠ N o Where M o Risky shares and N o Safety Shares in the Portfolio.
Let P ( x , y ; t ) be the joint probability generating function of P m , n ( t ) ;
where P ( x , y ; t ) = ∑ m = 0 ∞ ∑ n = 0 ∞ x m y n P m , n ( t )
Multiplying the Equations (2.4.1)-(2.4.4) with x m y n and summing overall m and n, we obtain
∑ m = 0 ∞ ∑ n = 0 ∞ x m y n P ′ m , n ( t ) = ∑ m = 0 ∞ ∑ n = 0 ∞ − [ l ( { a α i f r + ( 1 − a ) α i o r } + { b α i f s + ( 1 − b ) α i o s } ) ] x m y n P m , n ( t ) + ∑ m = 0 ∞ ∑ n = 0 ∞ − [ m ( { c α f r + ( 1 − c ) α o r } + { e τ f r + ( 1 − e ) τ o r } + { g δ f r + ( 1 − g ) δ o r } ) ] x m y n P m , n ( t ) + ∑ m = 0 ∞ ∑ n = 0 ∞ − [ n ( { d α f s + ( 1 − d ) α o s } + { f τ f s + ( 1 − f ) τ o s } + { h δ f s + ( 1 − h ) δ o s } ) ] x m y n P m , n ( t ) + ∑ m = 0 ∞ ∑ n = 0 ∞ [ l { b α i f s + ( 1 − b ) α i o s } + ( n − 1 ) { d α f s + ( 1 − d ) α o s } ] x m y n P m , n − 1 ( t ) + ∑ m = 0 ∞ ∑ n = 0 ∞ [ l { a α i f r + ( 1 − a ) α i o r } + ( m − 1 ) { c α f r + ( 1 − c ) α o r } ] x m y n P m − 1 , n ( t ) + ∑ m = 0 ∞ ∑ n = 0 ∞ [ ( n + 1 ) { h δ f s + ( 1 − h ) δ o s } ] x m y n P m , n + 1 ( t ) + ∑ m = 0 ∞ ∑ n = 0 ∞ [ ( m + 1 ) { g δ f r + ( 1 − g ) δ o r } ] x m y n P m + 1 , n ( t ) + ∑ m = 0 ∞ ∑ n = 0 ∞ [ ( n + 1 ) { f τ f s + ( 1 − f ) τ o s } ] x m y n P m − 1 , n + 1 ( t ) + ∑ m = 0 ∞ ∑ n = 0 ∞ [ ( m + 1 ) { e τ f r + ( 1 − e ) τ o r } ] x m y n P m + 1 , n − 1 ( t )
d P ( x , y ; t ) d t = [ { g δ f r + ( 1 − g ) δ o r } − ( { c α f r + ( 1 − c ) α o r } + { e τ f r + ( 1 − e ) τ o r } + { g δ f r + ( 1 − g ) δ o r } ) x + { e τ f r + ( 1 − e ) τ o r } y + { c α f r + ( 1 − c ) α o r } x 2 ] ∂ P ∂ x + [ { h δ f s + ( 1 − h ) δ o s } − ( { d α f s + ( 1 − d ) α o s } + { f τ f s + ( 1 − f ) τ o s } + { h δ f s + ( 1 − h ) δ o s } ) y + { f τ f s + ( 1 − f ) τ o s } x + { d α f s + ( 1 − d ) α o s } y 2 ] ∂ P ∂ y + [ l ( { a α i f r + ( 1 − a ) α i o r } ( x − 1 ) + l { b α i f s + ( 1 − b ) α i o s } ( y − 1 ) ] P ( x , y ; t )
We can obtain the characteristics of the model by using the joint C.G.F. of P m , n ( t ) as below
m 1 , 0 ( t ) = exp [ − { g δ f r + ( 1 − g ) δ o r } − { e τ f r + ( 1 − e ) τ o r } + { c α f r + ( 1 − c ) α o r } t ] M o (2.5.1)
where No is initial number of shares in the Risky Asset group
m 0 , 1 ( t ) = exp { − [ h δ f s + ( 1 − h ) δ o s ] − [ f τ f s + ( 1 − f ) τ o s ] + [ d α f s + ( 1 − d ) α o s ] } N o (2.5.2)
where Mo is initial number of shares in the Safety group.
m 2 , 0 ( t ) = { g δ f r + ( 1 − g ) δ o r } + { e τ f r + ( 1 − e ) τ o r } + { c α f r + ( 1 − c ) α o r } ∗ M o ∗ exp [ − { g δ f r + ( 1 − g ) δ o r } − { e τ f r + ( 1 − e ) τ o r } + { c α f r + ( 1 − c ) α o r } t ] ÷ [ − ( − { g δ f r + ( 1 − g ) δ o r } − { e τ f r + ( 1 − e ) τ o r } + { c α f r + ( 1 − c ) α o r } ) ] + exp [ 2 ( − { g δ f r + ( 1 − g ) δ o r } − { e τ f r + ( 1 − e ) τ o r } + { c α f r + ( 1 − c ) α o r } ) t ] (2.5.3)
m 0 , 2 ( t ) = [ { h δ f s + ( 1 − h ) δ o s } + { f τ f s + ( 1 − f ) τ o s } + { d α f s + ( 1 − d ) α o s } ] ∗ N o ∗ exp [ − { h δ f s + ( 1 − h ) δ o s } − { f τ f s + ( 1 − f ) τ o s } + { d α f s + ( 1 − d ) α o s } t ] ÷ [ − ( − { h δ f s + ( 1 − h ) δ o s } − { f τ f s + ( 1 − f ) τ o s } + { d α f s + ( 1 − d ) α o s } ) ] + exp [ 2 ( − { h δ f s + ( 1 − h ) δ o s } − { f τ f s + ( 1 − f ) τ o s } + { d α f s + ( 1 − d ) α o s } ) t ] (2.5.4)
m 1 , 1 ( t ) = exp [ − ( { g δ f r + ( 1 − g ) δ o r } + { h δ f s + ( 1 − h ) δ o s } + { e τ f r + ( 1 − e ) τ o r } + { f τ f s + ( 1 − f ) τ o s } − { c α f r + ( 1 − c ) α o r } − { d α f s + ( 1 − d ) α o s } ) t ] (2.5.5)